### Abstract

Let C_{n} be the set of Dyck paths of length n. In this paper, by a new auto- morphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set C_{n} is equidistributed with the statistic 'number of up steps at height h with h ≡ 0 (mod 3)'. Moreover, for m ≥ 3, we prove that the two statistics 'number of up steps at height h with h ≡ 0 (mod m)' and 'number of up steps at height h with h ≡ m - 1 (mod m)' on the set C_{n} are 'almost equidistributed'. Both results are proved combinatorially.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 2011 May 12 |

### Keywords

- Continued fraction
- Dyck path
- Exterior pair
- Ordered tree
- Planted tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

Eu, S-P., & Fu, T. S. (2011). Exterior pairs and up step statistics on Dyck paths.

*Electronic Journal of Combinatorics*,*18*(1).